Today’s surveys face a growing problem with increasing nonresponse.  The increase in nonresponse rate causes a need for better and more effective ways to reduce the nonresponse bias.  There are three major scientific orientation of today’s research dealing with nonresponse. One is examining the social factors, the second one studies different data collection methods and the third investigating the use of weights to adjust estimators for nonresponse.  We would like to contribute to the third orientation by evaluating estimators which use and adjust weights based on auxiliary variables to balance the survey nonresponse through simulations. For the simulation we use an artificial population consisting of 35455 participants from the Representativity Indicators for Survey Quality project. We model three nonresponse mechanisms (MCAR, MAR and MNAR) with three different coefficient of determination s between our study variable and the auxiliary variables and under three response rates resulting in 63 simulation scenarios. The scenarios are replicated 1000 times to acquire the results. We outline our findings and results for each estimator in all scenarios with the help of bias measures.

Šiame darbe nagrinėjami dviejų sumų santykio kalibruotieji įvertiniai, kuriuose panaudojama daugiau negu po vieną kiekvieno tyrimo kintamojo papildomąjį kintamąjį. Naudojant skirtingas kalibravimo lygtis ir atstumo funkcijas čia sukonstruoti šeši tokio tipo įvertiniai. Taikant Teiloro ištiesinimo, visrakčio ir atsitiktinių grupių metodus sukonstruoti keleto naujų santykio įvertinių dispersijos įvertiniai. Modeliuojant nauji santykio įvertiniai lyginami tarpusavyje bei su standartiniu ir A. Plikuso įvertiniais. Tiriama, kaip įvertiniu tikslumą įtakoja imties dydis ir laisvai pasirenkami svoriai, kai koreliacija tarp tyrimo ir papildomų kintamųjų yra stipri. Keičiant imties dydį pastebėjome, kad daugeliu atvejų nauji santykio įvertiniai yra tikslesni už A. Plikuso įvertini. Taip pat pastebėjome, kad parenkant skirtingus laisvai pasirenkamus svorius įvertinių tikslumas išlieka panašus. Atliekant sukonstruotų santykio įvertinių dispersijų įvertinių tikslumo tyrimą, kai keičiamas imties elementų skaičius, pastebėjome, kad visrakčio metodu gautieji dispersijos įvertiniai yra tikslesni už Teiloro ištiesinimo ar atsitiktinių grupių metodais gautus dispersijos įvertinius. Visi matematinio modeliavimo eksperimentai atlikti, naudojant matematinių uždavinių paketą MATLAB 7.10.0. / In this work we analyze the calibrated estimators of the ratio of two totals, which use more than one auxiliary variable for each study variable. Using different calibration equations and distance functions, we construct here six new estimators of the ratio. The estimators of the variance of some estimators of the ratio are constructed using Taylor linearization, jackknife and random groups methods. A simulation study is performed to compare new estimators of the ratio with the standard estimator and estimators, introduced by A. Plikusas. It is analyzed, how the characteristics of accuracy of estimators depend on the sample size and free additional weights when auxiliary variables are well correlated with study variables. The simulation results show that for some populations new estimators are more accurate than those, introduced by A. Plikusas. During the simulation, we observed, that the estimators of the variance of the estimators of the ratio that are constructed using jackknife method are more accurate than those that are constructed using Taylor linearization and random groups methods. The simulation results are obtained using the computer program that was made using the Language of Technical Computing MATLAB 7.10.0.

<p dir="ltr">This thesis presents advances in numerical methods for partial differential equations (PDEs) and optimization problems, with a focus on improving efficiency, stability, and accuracy across various applications. We begin by addressing 3D Poisson-type equations, developing a GPU-accelerated spectral-element method that utilizes the tensor product structure to achieve extremely fast performance. This approach enables solving problems with over one billion degrees of freedom in less than one second on modern GPUs, with applications to Schrödinger and Cahn<i>–</i>Hilliard equations demonstrated. Next, we focus on parabolic PDEs, specifically the Cahn<i>–</i>Hilliard equation with dynamical boundary conditions. We propose an efficient energy-stable numerical scheme using a unified framework to handle both Allen<i>–</i>Cahn and Cahn<i>–</i>Hilliard type boundary conditions. The scheme employs a scalar auxiliary variable (SAV) approach to achieve linear, second-order, and unconditionally energy stable properties. Shifting to a machine learning perspective for PDEs, we introduce an unsupervised learning-based numerical method for solving elliptic PDEs. This approach uses deep neural networks to approximate PDE solutions and employs least-squares functionals as loss functions, with a focus on first-order system least-squares formulations. In the realm of optimization, we present an efficient and robust SAV based algorithm for discrete gradient systems. This method modifies the standard SAV approach and incorporates relaxation and adaptive strategies to achieve fast convergence for minimization problems while maintaining unconditional energy stability. Finally, we address optimization in the context of machine learning by developing a structure-guided Gauss<i>–</i>Newton method for shallow ReLU neural network optimization. This approach exploits both the least-squares and neural network structures to create an efficient iterative solver, demonstrating superior performance on challenging function approximation problems. Throughout the thesis, we provide theoretical analysis, efficient numerical implementations, and extensive computational experiments to validate the proposed methods. </p>